Question

I'm having a lot of trouble with this problem:

Sketch the graph and show all local extrema and inflections.
f(x)= (x^(1/3)) ((x^2)-175)
I graphed the function on my graphing calculator and found the shape.
I also found the first derivative:
(7/3)(x^(4/3)) - (175/3)(x(-2/3))
But the number I found for x (when I set the deriv. to 0 to find the critical points) was 1.71, but this doesn't correspond to a min or a max. Then I plugged the equation of the first derivative in onto my calculator, and the corresponding y value of 1.71 was NOT zero.... furthermore, it seemed like there was a horizontal asymptote at 0, because the derivative never was 0.
I couldn't find an inflection point, because the second derivative never equaled zero.

Does this seem right-- that there is no local max, min, OR inflection point? If this is wrong, can you help me find the right values? I sketched the graph on my paper but I still haven't found the min, max, or inflection points.

Answer 1

I had the same first derivative as you did

y' = (7/3)(x^(4/3)) - (175/3)(x(-2/3))
= (1/3)x^(-2/3)[7x^2 - 175] by factoring
setting this equal to zero, ...
the first factor of (1/3)x^(-2/3) yields no solution but
7x^2-175=0 gives me
x = ± 5

my second derivative was (28/9)x^(1/3) + (350/9)x^(-5/3) which when set to zero has no answer.

so according to the math, there should be 2 points of max/min, but no point of inflection.
Does that fit in with your sketch?
I did not sketch the graph.